An Ecosystem Box Model for Estimating the Carrying Capacity of a Macrotidal Shellfish System

MARINE ECOLOGY PROGRESS SERIES Published December 1 Mar. Ecol. Prog. Ser.

An ecosystem box model for estimating the carrying capacity of a macrotidal shellfish system

'IFREMER, Unite de Recherche Ecosystemes Aquacoles, BP 133, F-17390 La Tremblade, France 'IFREMER, Direction de L'Environnement et de llAmenagement Littoral, Laboratoire de Chimie et Modelisation des Cycles Naturels, BP 70, F-29280 Plouzane, France

ABSTRACT: An ecosystem model is presented for estimating the shellfish carrying capacity of Marennes-Oleron Bay (France). It incorporates physical and biological processes: horizontal transport of suspended matter, feeding and growth of the cultivated oyster Crassostrea gigas and primary production. The precision and consistency of the model are tested by comparing simulations to observed data. The model smoothes both spatial and temporal variability of suspended-matter concentrations but reproduces mean levels and seasonal cycles with some accuracy. Carrying capacity of the shellfish system is assessed by computing the sensitivity of oyster growth to oyster abundance. Model results clearly inbcate a density dependence of oyster growth. When stock is adjusted from 20% to 200% of the present value, maximal dry weight of oysters shows a mean decrease of approximately 25%. The spatial heterogeneity of the growth response is a consequence of the low level of primary production within the shellfish area. The hydrodynan~icregime of the bay strongly controls the carrying capacity of the shellfish system: flushing time and available light energy are found to be 2 determining factors which prevent the phytoplankton from thriving in the bay; tidal currents ensure a fast renewal of food. Nevertheless, phytoplanktonic production accounts for a non-negligible part of food filtered by oysters and is identified as an Important food source when stock level is low. The validity of the model is h- ited mainly by the present description of the physical transport of suspended and deposited matter.

KEY WORDS: Oyster farming . Carrying capacity . Ecological box model . Crassostrea gigas. Stock sensitivity

INTRODUCTION This approach has the advantage of being simple and reliable. However, it provides a very restrictive picture Carrying capacity is a fundamental concept in shell- of the carrying capacity since it does not account for fish culture. Following Incze et al. (1981) and Rosen- potential variations of this capacity in space and time. berg & Loo (1983), it corresponds to the ability of the In addition, there are empirical models which depict system to support shellfish production. Mathematical some aspects of the interrelationship between shellfish tools are generally used to estimate carrying capacity. and food supply. The model developed by Incze et al. The complexity of the tools depends on the scientific (1981) estimates the optimal size of a mussel culture requirements, the ecosystem considered and the avail- from both seston concentration and water fluxes enter- ability of data. ing the system. In the model constructed by Wildish & Based on historical data, global models allow the Kristmanson (1979), the growth potential of suspen- overall production of a system to be represented as an sion-feeding macrobenthos is determined by the empirical function of the biomass (HBral et al. 1986). supply rate of ATP associated with advected seston. Smaal et al. (1986) correlate current velocity and shellfish biomass with seston depletion for different values 'Present address: CETIIS. 24 boulevard Paul Vaillant-Cou- of mussel bed length. In order to estimate the carrying turier, F-94200 Ivry-sur-Seine, France capacity of their system, Carvet & Mallet (1990) divide

0 Inter-Research 1994 Resale of full article notpermltted 118 Mar Ecol. Prog. Ser 115: 117-130, 1994

food supply, determined from measurements of tidal temporal framework. The goal is to establish relations exchanges and food concentration, by food demand, between oyster growth and biomass and also to point estimated from in situmeasurements of grazing. out the physical and biological processes which deter- Whatever their respective efficiency, the validity of mine the carrying capacity of the bay. these models is limited to restricted spatial and tempo- The study is composed of 2 parts. First, the consis- ral scales, as they model neither the impact of shellfish tency and precision of the model are assessed by com- culture on the overall dynamics of the system, nor the panng calculations to the data. Second, the model is regeneration of food within the shellfish system, nor used to quantlfy the sensitivity of oyster growth to oys- the spatial variability of both biological demand ter biomass. and physical characteristics of the system. When com- plex ecosystems are considered, a more explanatory, process-based model is needed (Heral et al. 1983a). MATERIAL AND METHODS The purpose of the present work is to refine the concept of carrying capacity by developing an ecosystem General description of the bay. Marennes-Oleron model which describes the physical and biological Bay is situated in southwest France. It is delimited by dynamics of a shellfish system. Oleron Island on its western side and by the continen- Macrotidal estuaries are favourable areas for shell- tal coast on its eastern side (Fig. 1). Water exchange fish culture. Being protected from the turbulence of the with the sea occurs in the north through the narrows of sea, such estuaries offer good trophic conditions due Antioche and in the south through the narrows of Mau- to strong tidal currents which ensure an intensive musson. The area of the bay is 200 km2 and its mean renewal of food within the area. Nevertheless, the depth is about 5 m, with an average of 10 m in the potential production of these areas is not unlimited. The case of Marennes-Oleron Bay, France, is a good illustration of this fact. Since its introduction into the bay in 1972, Japanese oyster Crassostrea gigas has shown a progressive reduction in its capacity for growth. The period of growth required for the oyster to reach a commercially valuable size has increased from 1.5 to 4 yr during the period 1972 to 1985 (Heral et al. 1986). At the same time, the mortality rate of the oyster has also increased. As shown by Heral et al. (1988), such physiological problems in C. gigas may have a trophic origin. By increasing the cultivated biomass, oyster-farmers may have reduced the food supply to the oysters. Because of the economic consequences of this situation, the problem of estimating the carrying capacity of Marennes-Oleron Bay has become a priority. Bacher (1989) has developed a first model for this which cou- ples the feeding behaviour of Crassostrea gigas and the physical transport of food. It reveals density dependence of oyster growth and strong intra-specific com- petition. However, the model is based on the assumption that renewal of food by primary production is negl~giblewhen compared to the inputs by tidal currents at the ocean boundary. On this basis, the 'stock/ growth' relationship computed by Bacher's model infers that sensitivity of the carrying capacity to primary production does not depend on stock level. This assumption is reasonable for small changes in stock level but may become unreasonable when large variations are considered. The present model is designed to incorporate feeding behaviour and growth of oysters, primary produc- Fig. 1 Marennes-Oleron Bay, France, showing the maln cir- tion and physical transport in the same spatial and culation features. Isobaths in meters Raillard & Menesguen: Box model of a shellfish ecosystem 119

north. The bay is a dynamic environment with a tidal (L2). Elementary cells of the hydrodynamic model range up to 5 m. The local hydrodynamic regime is were aggregated into boxes in order to match as far as regulated mainly by tidal currents, resulting in a possible the physical and biological gradients in the southward residual transport of the water masses. Res- bay including those of: current velocity and direction, idence time of a water parcel in the bay is between salinity (due to Charente Estuary), and oyster distribu- 5 and 10 d, depending on tide and meteorological con- tion (flooding areas/gullies). The amplitude of the tidal ditions. The River Charente is the major source of excursion, which is responsible for water mixing, freshwater input into the bay. Although river flow is determines the mean geographic extent of the boxes. negligible (1 %) in comparison to oceanic inflow (99 X), The segmentation of Marennes-Oleron Bay is shown in it can be an important source of nutrients, thus increas- Fig. 2. For box volume computation, only marine area ing primary production (Ravail et al. 1988). During is taken into account. periods of flooding, the waters of Gironde enter the When interaction with biological processes is absent, bay (Dechambenoy et al. 1977) and have been shown the concentration of seston Ci in Box i over time t is to increase nutrient levels significantly. calculated by the following equation: Crassostrea gigas is cultivated in the bay. Its estimated biomass is approximately l l0 000 t and annual biological production can rise up to 35 000 t (A. Bodoy pers, comm.). This latter value marks Marennes- where n is the total number of boxes, V, is the volume Oleron Bay as one of the most productive areas for oys- of Box i, A, is the advective flow coming out of Box i, ters in the northeast Atlantic. Ecosystem model. Physical sub-model: A hydrodynamic model was developed (Anon. 1979) to calculate vertically averaged current velocity and direction throughout the bay. Considering the vigorous hydrodynamic regime and the shallow bathymetry of the bay, it can be assumed that the water column 1s vertically homogeneous. In this model, tidal forcing is the only factor determining water circulation. Ecosystem models rarely require the spatial and temporal resolution needed for accurate hydrodynamic calculations (Bird & Hall 1988). Following Chen & Smith (1979), Radford & Joint (1980), Lindeboom et al. (1988) and Bacher (1989), the hydrodynamic outputs of the physical model were averaged over time and space, resulting in an advection-dispersion 'box model' that drives the horizontal transport of dissolved and particulate matter within Marennes-Oleron Bay. The procedure for calculating spatial and temporal distribution of the parameters of the advection-dispersion model using the outputs of the hydrodynamic model is detailed in Bacher (1989). Fluxes between contiguous boxes are integrated over the tidal period (12 h 25 min), so only tidal residual circulation is considered, and it is assumed that biological processes are not affected by higher frequency variations of environmental parameters. The alternation between spring and neap tides is simulated by linear interpolation between 'spring tide' and 'neap tide' velocity fields alternating every 7.3 d. The geographic position of the model is determined so that one can assume that the calculations will not influence the boundary conditions. Three boundaries are considered: the northern and southern boundaries Fig 2. Division of Marennes-Oleron Bay into model compart- (Ll) receive the same oceanic inputs, and the River ments (i.e boxes); the limits of the box model are: L1,oceanic Charente enters the bay through the eastern boundary limlts; L2, river limit 120 Mar. Ecol. Prog. Ser. 115: 117-130, 1994

M~neral sesron

Nutr~ents plankron

TIME (DAYS)

Fig. 3. Comparison of observed (-) and simulated (...) salinity in Box 14

A,. is the advective flow entering from Box j into Box i, and D,, is the dispersive flow between Box i and Box j. Depos~ ted Dispersive flows Dd are used to calibrate the trans- SEDIMENT port box model by fitting simulated salinity to observed c3 values collected in the middle of the bay during 1 yr. o4 Satisfactory agreement between simulated and observed data can be obtained (Fig. 3). Fig. 4. Process flow diagram for the model. I,,,: light at depth z The transport equation is solved by a first upward and time t; T: temperature; POM: particulate organic matter differencing scheme (Bird & Hall 1988). Biological sub-model: Because inorganic N is known to be the primary limiting nutrient for phyto- Japanese oysters (Bayne et al. 1976, Soniat et al. 1984). planktonic growth in Marennes-Oleron Bay (Heral et Faeces, phytoplankton and SPOM supply the benthic al. 1983b), the model simulates the nitrogen cycle in POM (Kautsky & Evans 1987). As a first approximation, the water and the sediment. State variables and their we assume that pseudofaeces presence does not affect relationships are shown in Fig. 4. The system of differ- the accessibility of food to oysters (Newel1 & Jordan ential equations and the functions used in the model 1983). The demographic structure of the oyster popula- are detailed in Tables 1 & 2 respectively. tion is represented by 2 age classes: the first class Driving variables are temperature (T), intensity of includes the 1 yr old oysters, and the second includes photosynthetically active radiation (I) and sea state 2 and 3 yr old oysters. The temporal evolution of the (Turb). The seasonal cycle of temperature is modelled numbers of the 2 age classes is driven by a first order by adjusting a sinusoidal curve to the data collected in law (Bacher 1989). the bay (see Table 2). Solar radiation at the sea sur- For the other components of the ecosystem, the face is derived from daily insolation observations equations of the biological sub-model are presented in using the Brock (1981) method. Sea state is a syn- Table 2. The effect of temperature on various rates is thetic variable which drives vertical exchanges of par- considered as being exponential over the usual range ticulate matter, with a trend to erosion in winter and of sea temperature observed in Marennes-Oleron Bay. to sedimentation in summer (Sornin 1981). Accord- Phytoplanktonic growth is reduced by nutrient limita- ingly, sea state is described by a sinusoidal function of tion according to a Michaelis-Menten hyperbola and time (Table 2). by light energy according to the Steele (1962) function. Processes concerning feeding behaviour and growth The extinction coefficient is linearly dependent on ses- of individual oysters have been detailed by Raillard ton concentration (Cloern 1987). In our model, phyto- et al. (1993). Suspended particulate organic matter plankton settling on the bottom is considered a natural (SPOM) and phytoplankton are the food source of mortality process, with a rate linearly depending on Raillard & Menesguen: Box model of a shellfish ecosystem 121

the nutrient limitation factor (Eppley et al. 1967). Graz- Daily concentrations at the box boundaries for all the ing of phytoplankton and detrital matter by zooplank- pelagic variables are calculated by a linear interpola- ton is modelled by an Ivlev curve; grazing occurs tion of monthly averages [see Bacher (1989) for the above a threshold concentration and remains null sampling strategy used in Marennes-Oleron Bay]. As a under this threshold (Parsons et al. 1967, Frost 1972). result, the annual cycle at the boundaries is free of Zooplanktonic assimilation efficiency is an exponen- accidental perturbations (Fig. 5). tially decreasing function of the weight of ingested As the model was revealed to be insensitive to initial organic matter (Gaudy 1974). Zooplankton excretion values of pelagic variables (which could be determined rate is the higher of 2 terms: a constant fraction of the from in situ measurements), in a first approximation, assimilated food (Corner et al. 1967) or basal excretion initial values of benthic nitrogen stocks (for which rate as a function of temperature. Nitrogen remineral- there are no data available) are set equal to zero. The ization is governed by a first order law. Sedimentation initial number of oysters in each box is determined and resuspension, which link detrital organic matter in according to an estimate based on a sampling survey the water and the sediment, are driven by the external in 1984 (Bacher et al. 1986). Boxes 10, 11, 13, 14 and 15 variable Turb. Values of the parameters were set contain oysters of both age classes. according to values currently used in similar models The differential equations for the coupled physical (Table 3). and biological sub-models are integrated using a Coupling the physical and the biological sub- fourth order Runge-Kutta time step varying method. models: The transport model does not account for spe- Simulations: For the pelagic variables, observed val- cific meteorological processes such as winds and ues are presented using the same method as that used storms (Bacher 1989). Accordingly, the ecological for the boundaries. Only 2 years of data are available model uses data corresponding to an average year. for oyster weight. Assuming that the oyster growth

Table 1. System of differential equations used in the model. Hwat: average water depth of the box. Other variables defined in Table 2

Dissolved mineral nitrogen (pmol N 1-l) I dNMIN = - Pgrowth, NPHY + Nremin. NDET + Zexcr NZ00 - dt Phytoplankton nitrogen (pmol N I-') --dNPHY - (Pgrowth - Pmort).NPHY - Pgraz.NZO0 - Tingphy dt

Zooplankton nitrogen (pmol N 1-l)

--dNZOO - (pgraz-asszoo - Zexcr - Zmort) . NZOO dt Detrital nitrogen (pmol N 1-l) dNDET NBEN Nsedim ----- = Pmort - NPHY + (Zegest +Zmort) - NZOO + Nresusp . - dt -Hwat [- Hwat Benthic nitrogen (mm01 N m-')

--dNBEN - Nsedirn - NDET - Nresusp. NBEN + (Tfecphy + Tfecdet) - Hwat dt Oyster dry weight (J ind.-l) dOys = Sfg - Spawn - dt Oyster number for the Mh age class

Mineral seston (mg dry wt I-') 122 Mar. Ecol. Prog. Ser. 115: 117-130, 1994

Table 2. Equations of the biological submodel. VolB box volume. Other variables defined in Tables 1 P1 3

Quantity Meaning Formula

Pgrowth Phytoplankton growth rate Pmax.f (11.f (T1.f (NI f (T) Temperature effect ekl T 24 rmax fi-k\ j l"",dzdt f (1) Light effect on phytoplankton Fe. O zrnm OPt IZ,t Light at depth z and time t Isurf.t.e-kl' kl Light extinction coefficient akt.SES + akO NMIN f (NI Nitrogen limitation NMIN +kN

Pmort Phytoplankton mortality rate mnphy.f (NI + mxphy.11 - f (NI1 Pgraz Zooplankton grazlng rate rmax.f (T).( 1 - e-" mma.Y(O."PHY-PoI I asszoo Zooplankton assimilation efficiency aSSmaX.e-ka.Pgraz Zegest Zooplankton egestion rate (l - asszoo).Pgraz Zexcr Zooplankton excrehon rate max[Pgraz~asszoo~excrphy,excrzoo.f (T)] Zmort Zooplankton mortality rate m2oo.f (T)

"c Stox 1 Ting[phy,det] Oyster ingestion E I~~IPHY,DETI~, k=~!=I -VolB Ing[phy,detIi, Inhvidual oyster ingestion F.N[phy,det].(l- PF) F Individual oyster filtration rate ~~~~.~ldrnm(O.Tses -SES).wbf W Oyster dry welght in grams OYS.P~ PF Proportion rejected as pseudo-faeces p~~.(l-~kp~m1310.=t --ct)) + (l-p~~).(lekp2 ~IZ~O,C?-CII 1

F [SES + (NPHY+ NDET) - pN] Ct Standardized oyster consumption Wbl Tfec[phy,det] Oyster egestion Ting[phy,det].(l- assoys) assoys Oyster assimilation efficiency aet,T + aeO Sfg Individual oyster 'scope for growth' assoys.(Ingphyb.ephy + Ingdet,.edet) - R.eOz R Individual oyster respiration (art.T + arO),CVbr Spawn lnhvidual oyster spawnlng (ap.Oy~~P)~[~p Nremin Orga~ucnitrogen minerallzatlon m1nazote.f (T) Nsedim Organic nitrogen sedimentation rate sedmin.Turb + sedmax.(l - Turb) Nresusp Benthic nitrogen resuspenslon resmln.Turb + resmax.(l - Turb)

( 2iT Turb Sea state l+sn --(-140)) \ 365 2 t Jullan days

curve follows a general pattern that does not change sirnulations were performed, using initial values of from one year to another, these data may be used to oyster abundance scaled by a factor varying from 0.2 to test the ability of the model to simulate oyster weight 2, in each box and for both age classes. The maxlmal evolution. dry weight reached by oysters in the different boxes To study the relationship between oyster growth and was then plotted aga~nstthe corresponding factor- the total biomass under cultivation, several 1 yr long adjusted initlal stocks. Raillard & Menesguen: Box model of a shellfish ecosystem 123

RESULTS have little influence on phytoplanktonic growth. Dur- ing the rest of the year, calculated values stayed closer Standard simulation to the measured ones. In the middle of the bay (Box 14), simulation of nitrate represented with some Unfortunately, the model failed to simulate accu- accuracy the overall average concentration and its sea- rately the wintering levels of nitrate in Box 8 (Fig. 6), sonal cycle (Fig. 6). However, this discrepancy between model and data Simulation of mineral seston in Boxes 8 and 14 fol- does not call into question the consistency of the bio- lowed the mean trend of observed concentrations. logical sub-model, since winter variations of nitrate Simulated changes in phytoplankton biomass in Box 8

Table 3. Values of the parameters used in the biological sub-model

Parameter Meaning Value

Phytoplankton pmax Maximal growth rate at O°C 0.5 d-' kN Half saturation constant for N hmitation 1.5 pm01 1-' N Iopt Optimal light intens~ty 70 W m-' akt Slope of the kl curve vs SES 0.06 m2 g-' dry wt a kO Intercept of the W curve vs SES 0.17 m-' kt Temperature coefficient 0.07 "C-' mxphy Maximal mortality rate 0.1 d-' mnphy Minimal mortality rate 0.01 d-' ephy Energy equivalent 2.86 J pmol-' N Zooplankton rmax Maximal growth rate at O°C kz Ivlev's constant ka Assimilation efficiency exponent PO Ivlev's grazing threshold assmax Maximal assimilation efficiency excrzoo Excretion rate at 0°C excrphy Excreted ratio of the assimilated food mzoo Mortality rate at 0°C Nitrogen minazote ~Vineralizationrate at 0°C 0.05 d-' sedmin Minimal sedimentation velocity of detrital matter l m d-' sedmax Maximal sedimentation velocity of detrital matter 2.5 m d-' resmin Minimal resuspension rate of detntal matter 0.02 d.' resmax Maximal resuspension rate of detntal matter 0.2 d-' P N pm01 N to mg dry wt conversion 0.4 mg dry wt pmol-' N edet Energy equivalent 1.68 J pmol-l N Oyster Fmax Maximal filtration rate 48 1 d-' g-' dry wt kf Filtration exponent for clogging 0.07 Tses Clogging threshold 200 mg 1-' bf Allometnc exponent of filtration 0.4 PFo Pseudo-faeces production step 0.4 kpl, kp2 Pseudo-faeces exponents 0.15. 0.01 cl, c2 Pseudo-faeces thresholds 120, 2400 mg dry wt d-' g-' dry wt aet Slope of assimilation curve vs temperature 0.033 "C-' aeO Intercept of assimilation curve vs temperature 0.033 art Slope of respiration curve vs temperature 0.768 m1 O2 d-' g-' dry wt "C-' arO Intercept of respiration curve vs temperature -0.528 m1 O2 d-' g-' dry wt br Allometric exponent of respiration 0.7 aP Proportionality constant of spawning bp Allometnc exponent of spawning dp Spawning period Hm~rt,,~ Mortality rate for the first and second age class e02 Energetic conversion of oxygen P1 Mass conversion of energy 0.05 g dry itkJ-' nc Number of age classes 2 124 Mar. Ecol. Prog. Ser. 115: 117-130, 1994

7 . . L 10k' : NPHY 8 8; E 3 61 C.

M' JS'N' Months Months Months

\ NDET ',

4L, v J' ' M M- Months Months ll - ' " ' S N 7-. 80-' Months E60- : SES SES ii

.- I - O-m' O-m' . J. . S. N. - Months t 0-L- ---. Fig. 5. Temporal distribution of the pelagic variables at the boundaries of the box J M M model. L1: oceanic limits; L2: river limit MonthsJSN'- corresponded reasonably well to the data in winter and determined more by seasonal changes in the environ- spring but did not reproduce the secondary bloom in ment than by the springheap variability, which was summer. In Box 14, calculated phytoplankton biomass poorly simulated by the model. showed a satisfactory agreement with the mean observed values. However, the model slightly underestimated concentrations during winter and spring Sensitivity of oyster growth to oyster stock blooms and predicted a high value in summer. In the estuary area (Box 6), the model largely underestimated Results indicated a global influence of stock level on the phytoplankton biomass throughout the year oyster growth. When the oyster stock was divided by 5, (Fig. 6). the maximal dry weight of oysters was increased on The springheap tidal variability observed for the average by 14 9i,;when the oyster stock was multiplied pelagic components was poorly simulated by the by 2, the maximal dry weight was reduced by 12%. model. If one omits the simulation of nitrate in Box 14, Another global feature of this result is the non-linear the model either smoothed the oscillations (see mineral nature of the growth sensitivity to stock level. This seston in the center of the bay, Box 14) or did not simu- form of growth density dependence has been observed late them at all (see phytoplankton biomasses in Boxes for historical data of Marennes-Oleron Bay (Heral et al. 6 and 14, or nitrate, phytoplankton and mineral seston 1986). in Box 8). Oyster growth sensitivity was also dependent on the The simulated oyster growth curve (Fig. 7) followed geographical location of the oysters. As shown in a pattern similar to the observed one: maximal growth Fig. 8, the growth of oysters located in the middle and in spring, stabilization during winter and autumn and a south of the bay (Boxes 13, 14 and 15) appears to be large loss in summer for the second age class due to more sensitive to stock level than the growth of oys- spawning. This indicates clearly that oyster growth is ters located in the north (Boxes 10 and 11). Because of Raillard & Menesguen: Box model of a shellfish ecosystem 125

Box 6 Box 8 Box 14 Charente estuary Boyard Chapus

--- 60-' ' ' ' ''-l ' t 'l r--+-----T

I 0 , , , , b..> , , , ,l JFMAMJ J ASOND JFMAMJ J ASOND Months Months Months

-4 500 -4

OF.. , . . . . , . . , .2 , o;...., ..,,,... ,: JFMAMJ J ASOND JFMAMJ J ASOND JFMAMJ J ASOND Months Months Months

JFMAMJ JASOND Months Months Months

Fig. 6. Simulated (-) and observed l.--)pelagic variables in Boxes 8. 6 and 14 the north-south flux of food through the bay, the northern boxes (10 and 11) provided the best oyster growth of oysters in the central area (Boxes 13 and 14) growth for large values of stock (>loo%). is determined not only by the local stock of oysters, Thus, the total oyster production of the bay depends but also by those oysters found in the northern area on 2 inversely correlated parameters: initial oyster (Boxes 10 and 1l), which previously deplete the food. density, and individual growth performance. When the The lower sensitivity of oysters located further south initial stock is increased, production tends to increase, (Box 15), compared to those in Boxes 13 and 14, is ac- whereas growth slows down. Ultimately, the negative tually due to the proximity of the southern oceanic influence of the increasing stock on individual growth boundary. The food supply coming from this bound- leads to a relative stabilization of total production. ary counterbalances the depletion of food coming Total production is reduced by 44 % when the oyster from the northern area. stock is divided by 2 (relative to the present situation) The variation of total oyster stock modified the gradi- yet is increased by only 28% when the stock level is ent of carrying capacity among the different boxes. multiplied by 1.5. So, the relative gain in production When stocks were low (

I, 2.0 - 0 1 2nd age class i 1.9 16 1 - BOX 18 .- BOX l4

J - 1 1st age class 2 1.6 !- n - z ;

J FMAMJ JASOND AMOUNT OF OYSTERS (%) MONTHS Fig. 8. Sensltivlty of oyster lnd~vidualgrowth to oyster stock. The stock is expressed as a percentage of the nominal value Fig. 7. Simulated (-1 and observed (0)indiv~dual growth of oysters in Box 14

DISCUSSION calculations for different boxes (Table 4), the flushing time of the water masses IS generally shorter than the The fact that the model reproduces the magnitude doubling time of phytoplankton (between 0.5 and 9 d), and the shape of distribution of the various constituent preventing significant accumulation of phytoplank- parameters indicates that relevant processes are tonic new material within the box volume (Officer accounted for (O'Connor 1981). Consequently, this 1980).The interaction between flushing time and bio- suggests that the present model may be used to make logical processes has been previously described by Es assumptions about the factors that regulate carrying & Ruardij (1982) and Helder et al. (1983), by integrat- capacity of the shellfish system. ing this time scale in the computation of oxygen con- The results of the sensitivity analysis can be inter- sumption and nutrient production. The overriding preted in terms of dynamic food supply. The north- effect of light limitation on photosynthesis when com- south gradient of growth sensitivity in response to vari- pared to nitrate limitation is clearly illustrated by the ation of stock reveals the overriding effect of hydrody- growth limitation curves displayed in Fig. 9. In Box 8, namic processes on the renewal of food within the bay. the greater water depth (see Table 4) together with the Production of phytoplankton in the shellfish area is not assumption of vertical homogeneity of the water col- actually sufficient to balance the depletion induced by umn explain the strong limitation of phytoplanktonic the feed~ngactivity of oysters. The flushing time of growth by light energy. In contrast, in Boxes 6 and 14, water masses and the available light energy in the as shown by the springheap tidal variability of light water column are found to be 2 factors that prevent hmltation, the level of mineral seston controls light llm- phytoplankton from thriving in the bay. As shown by itation. Ra~llard& Menesguen: Box model of a shellfish ecosystem 127

Table 4. Physical and biological characteristics of the boxes in Marennes-OleronBay

Box 6 Box 8 Box 10 Box 11 Box 13 Box 14 Box 15

Volume (X 1O6 m') 26.55 149.14 110.88 79.78 4.46 79.10 93.43 Flushing time, min.-max. (d) 0.4-0.71 0.22-0.52 0.24-0.60 0.15-0.41 0.16-0.36 0.25-0.49 0.36-0.52 Number of oysters in: First age class (xlOh) 0 0 1.27 314.77 8 51 306.55 124.12 Second age class (x106) 0 0 15.81 220.03 17.69 626.94 496.49 Average depth (m) 2.53 11.32 5.38 4.31 0.92 2.65 3.57

The present model differs from the model developed between model and data have also shown significant by Bacher (1989) mainly by including phytoplanktonic discrepancies. The explanatory and predictive value of growth. The sensitivity analysis computed by the 2 the model may be assessed by analysing its various models shows 2 major differences. First, the range of faults (Baretta & Ruardij 1988). 'stock vs growth' curves computed for the different boxes is narrower when predicted by our model. From this observation, it can be asserted that the food supply BOX 8 arising from primary production partly balances the north-south depletion due to oyster feeding activity and subsequently smoothes the spatial heterogeneity of trophic capacity. Second, Bacher's model does not predict a change in the spatial gradient of oyster growth performance. In our model, oyster growth in Boxes 13,14 and 15 is better than in Boxes 10 and 11 in the case of low stock levels, whereas it is the contrary when stock levels are high. This may be due to the lower total filtering action of oysters situated north- MONTHS ward when stock levels are low, which allows the local production of phytoplankton to be sufficient for oyster production. Considering the sensitivity of the carrying capacity to primary production, the present model suggests, especially for large variations of oyster stock, that more confidence can be placed in predictions when the nutrient/phytoplankton system is simulated. In order to further illustrate the impact of oysters on food supply, the annual distribution of phytoplankton biomass computed for 2 levels of stock is plotted in Fig. 10. The computed decrease in phytoplankton bio- MONTHS mass when the number of oysters is trebled (i.e. from BOX 14 50 % to 150 % of the present value) indicates that molluscs do not strongly control phytoplankton biomass in Marennes-Oleron, which is contrary to what was observed in San Francisco Bay (Cloern 1982) or in the Oostershelde (Smaal et al. 1986). This model result can probably be related to the flushing time within the bay. The low residence time, which ensures rapid renewal of the food supply, is responsible for the high carrying capacity predicted by the model.

The appraisal of the shellfish system as suggested by JMMJSN the above interpretations of the model results naturally MONTHS depends on model reliability. Even if the present Fig. 9. Computed (dimensionless)limitation factors of phyto- model has produced realistic trends of the evolution planktonic growth by light availability and nutrient con- of several ecosystem components, the comparisons centration 128 Mar. Ecol. Prog. Ser.

Springheap tidal variability of particulate matter is sensitive to the variability of the driving parameters, one of the weakest points of the present model. The even if the seasonal trend of the evolution is con- simulated springheap tidal variability is actually due served. Furthermore, the influence of the vertical to differences in mixing between ocean and bay waters exchanges on suspended matter concentration is dunng neap and spring tide (Raillard 1991). However, dependent on both the bathymetry and the current considering the shallow depth and the vigorous hydro- velocity within the bay. This probably leads to in- dynamic regime of the bay, vertical exchanges be- creased spatial heterogeneity of food concentration. tween sediment and water may also control the con- We suggest that a more realistic picture of the physical centration of particulate matter. During neap tide, the and biological dynamics of the shellfish system could concentration would be increased because of low cur- be achieved with an explicit simulation of the erosion rent speed, but during spring tide the high current and sedimentation processes. velocity would enhance resuspension of particulate The overall underestimation of phytoplankton bio- matter. mass in the estuary (Box 6) can be explained by a lack From a modelling point of view, the deposition of of precision in the modelling of phytoplankton dynam- particulate matter would not largely affect the com- ics. Following Ravail et al. (1988), it can be assumed puted food supply. Even if the process can be supposed that (1) local primary production is actually enhanced to increase the residence time of particulate matter in by a specific species of phytoplankton that is well the bay (and hence the food supply), it also reduces, in adapted to low light levels, and (2) inputs of terrige- compensation, the period of food availability for oys- nous chlorophyll coming from the River Charente ters (the deposited food is not available for oysters). account for a non-negligible part of the phytoplankton However, the carrying capacity of the bay could be dif- biomass within the estuary. Nevertheless, considering ferently evaluated if the food is in fact fluctuating as that the Box 6 has a relatively small volume (Table 4) observed. Being generally non-linear, the mathemati- and that it does not contain oysters, it can be assumed cal expressions describing biological processes are that the incorrect estimation of the phytoplankton biomass in this box does not severely bias the overall prediction of the carrying capacity. -781'- Overestimation of the spatial homogeneity of the dis- -Z , BOX14 tribution of the pelagic components also has a numeri- 0 a - STOCK x 1.5 7 cal origin. The first upward differencing scheme used k6- STOCK X 8.5 4 - here to solve the transport equation has the advantage of being explicit, forward in time, and therefore very fast. However, the algorithm is accurate only to the first order in time and space and, therefore, produces a strong numerical diffusion (Bott 1989), resulting in both overestimation of the water mixing between the different boxes and underestimation of the flushing time (Bird & Hall 1988). The effective interactions 1 LLi JMMJSN between biological and physical processes are likely to MONTHS be biased by such a numerical defect. Globally, the F - reduction of the flushing time tends to cause an under- 28; " " ' 4 estimation of the influence of biological processes on the temporal evolution of pelagic variables within the t, - bay. However, the 2 processes that control food supply, -z. 6- - - STOCK x 0.5 ,'S . i.e. primary production and physical transport, are ; '%L inversely influenced by the flushing time: the former is decreased when flushing t~meincreases, whereas the latter is increased. Since the 2 effects can balance each other, the underestimation of flushing time may have Little influence on the estimation of the bay's carrying capacity. Nevertheless, the numerical scheme also 0-, masks the spatial heterogeneity of the system and sub- JMMJSN sequently limits its validity. MONTHS In order to reduce numerical diffusion, Bird & Hall Fig. 10. Temporal distribution of phytoplankton biomass for (1988) and Shanahan & Harleman (1984) suggest opti- 2 levels of oyster stock mizing the 'current number' by adjusting space and Raillard & M6nesguen: Box model of a shellfish ecosysteln 129

time steps of the system. But the differences between thelr ecology and physiology. IBP No. 10, Cambridge the volumes of the boxes make this solution inapplica- Univ. Press, Cambridge, p. 261-291 ble to the present case. It is also possible to integrate Bird. S., Hall, R.(1988). Coupling hydrodynamics to a multiple- box water quality model. Tech. Rep. EL-88-7, Waterways the biological laws inside the spatial and temporal Experiment Station, Corps of Engineers, Vicksburg, MS structure of the hydrodynan~icmodels. In such models, Bott, A. (1989).A pos~tivedef~nlte advection scheme obtained advection and dispersion of suspended matter are by nonlinear renormalization of the advective fluxes. Mon. computed by efficient algorithms that significantly Weather Rev. 117- 1006-1015 Brock. T D. (1981). Calculating solar radiation for ecological reduce numerical dispersion (Bott 1989). Even if one studies. Ecol. Model. 14: 1-19 ignores the computation cost of such an approach, we Carvet, C E. A., Mallet, A. L. (1990) Estimating the carrying think there are 2 other reasons to be doubtful about its capaclty of a coastal Inlet for mussel culture. Aquaculture relevance: (1) the validity of the biological laws is not 88: 39-53 Chen, C. W., Smith. D. J. (1979). Preliminary insight into a independent of the spatial and temporal context; (2) three-dimensionnal ecological-hydrodynamic model. In: the combination of the non-linearities of both hydrody- Scavia, D., Robertson, A. (eds.) Perspectives on lake eco- namic and biological models could lead to a numeri- system modelling Ann Arbor Pub1 Co., Ann Arbor, hd1, cally very unstable and consequently unreliable sys- p. 249-279 tem. We think that even if the box structure of the Cloern, J. E. (1982). Does the benthos control phytoplankton biomass in south San Francisco Bay? Mar. Ecol. Prog. Ser. ecosystem models drives to some extent the responses 9: 191-202. of the model, it is nevertheless a framework that con- Cloern, J. E. (1987). 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This article was submitted to the editor Manuscript first received: July 23, 1993 Revised version accepted: August 19, 1994